Well first your going to need a power source (battery). I assume you want this to last for a game or two of golf before you want to charge it again. A lithium ion or small sealed lead acid battery (more weight though) should do the trick.

I highly recommend differential (tank style) drive with a off road caster (for all those times you hit it in the rough )

In order to drive and control the motors, you'll need a motor driver. Check out the Sabertooth from Dimension engineering. If that's too expensive, there are plenty of other ones cheaper. The Sabertooth is high quality though, fairly easy to configure, and there is plenty of documentation on it.

For MCUs, I recommend using an Axon or Arduino. There is a ton of documentation on both.

If you want remote control, just search "rc transmitter and receiver" on google and you should find many options. Plus the Sabertooth has an rc mode that is really easy to set up and configure.

If you are going to use that as a base I would use the battery that it comes with. As far as what electronics you need depends on what you intend to do. If you simply want it to be teleoperated or if you want it to move buy itself requires completely different electronics.

Arduinos are all the rage. Very easy to program using C and can interface with multiple motors and sensors. But have a look at Raspberry Pi - this is a full on single board computer which you can interface with an arduino and have even more functionality for your Bot. For controlling high power (amp/voltage) motors from the arduino, you will need some motor drivers which take input signals from the arduino at 5V and can control motors which pull upto 30Amps at 18V. These controllers can be found on ebay or Element14 (for example).